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Local and global spatio-temporal entropy indices basedon distance- ratios and co-occurrences distributionsDidier G. Leibovici, Christophe Claramunt, Damien Le Guyader, David

Brosset

To cite this version:Didier G. Leibovici, Christophe Claramunt, Damien Le Guyader, David Brosset. Local andglobal spatio-temporal entropy indices based on distance- ratios and co-occurrences distribu-tions. International Journal of Geographical Information Science, 2014, 28 (5), pp.1061-1084.<10.1080/13658816.2013.871284>. <hal-01208089>

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Didier G. LEIBOVICI, Christophe CLARAMUNT, Damien LE GUYADER, David BROSSET - Localand global spatio-temporal entropy indices based on distance- ratios and co-occurrencesdistributions - International Journal of Geographical Information Science - Vol. 28, n°5, p.1061-1084 - 2014

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Local and global spatio-temporal entropy indices based ondistance-ratios and co-occurrences distributions

Didier G. Leibovicia*, Christophe Claramuntb, Damien Le Guyaderc and David Brossetb

aNottingham Geospatial Institute, University of Nottingham, Nottingham, UK; bNaval AcademyResearch Institute, Brest Naval, France; cUniversity of Bretagne Occidentale, UMR LETG, Géomer,

IUEM, Plouzané, France

When it comes to characterize the distribution of ‘things’ observed spatially andidentified by their geometries and attributes, the Shannon entropy has been widelyused in different domains such as ecology, regional sciences, epidemiology and imageanalysis. In particular, recent research has taken into account the spatial patternsderived from topological and metric properties in order to propose extensions to themeasure of entropy. Based on two different approaches using either distance-ratios orco-occurrences of observed classes, the research developed in this paper introducesseveral new indices and explores their extensions to the spatio-temporal domainswhich are derived whilst investigating further their application as global and localindices. Using a multiplicative space-time integration approach either at a macro ormicro-level, the approach leads to a series of spatio-temporal entropy indices includingfrom combining co-occurrence and distances-ratios approaches. The framework devel-oped is complementary to the spatio-temporal clustering problem, introducing a morespatial and spatio-temporal structuring perspective using several indices characterizingthe distribution of several class instances in space and time. The whole approach is firstillustrated on simulated data evolutions of three classes over seven time stamps.Preliminary results are discussed for a study of conflicting maritime activities in theBay of Brest where the objective is to explore the spatio-temporal patterns exhibited bya categorical variable with six classes, each representing a conflict between twomaritime activities.

Keywords: information theory; entropy; spatio-temporal entropy; co-occurrence data;nearest neighbor; spatial structuring; point pattern analysis

1. Introduction

The concept and measure of entropy as initially introduced by Shannon in his seminaltheory of information has been long applied to many scientific domains to qualify thedistribution of ‘things’ in space (Shannon 1948, Shannon and Weaver 1949). Shannon’sentropy is a mathematical index that measures diversity in categorical data. It is moreformally given by

HðCÞ ¼ �KX

cpc logðpcÞ (1)

where pc is defined as the proportion of entities of the class c of the categorical variable Cwith |C| classes (C being one attribute of the objects observed) and K is a positive

*Corresponding author. Email: didier.leibovici@nottingham.ac.uk

constant. The entropy H is a positive value, and it is bounded by the unit interval when Kis chosen as the inverse of the maximum entropy among discrete distributions with thesame number of classes. This maximum value, log(|C|), is reached when the distributionof classes is uniform ð"c; pc ¼ nc=N ¼ 1=jCjÞ, which increases with the number ofclasses, so that H with this normalizing factor describes the structuring of the distributionas departing from the uniform distribution.

Since the seminal contribution of Shannon the measure of entropy has been widelyapplied in various domains including for spatial and geographical data, such as in regionalsciences, image analysis, ecology and social sciences, see some review in Li and Huang(2002), Claramunt (2005), Leibovici (2009) and Batty (2010). In ecological and environ-mental studies (Margalef 1958, Menhinick 1964, McIntosh 1967, Hurlbert 1971, Gonzalezand Chaneton 2002), directly applied to the spatial data, the measure of entropy has beenused to evaluate the fragmentation and spatial heterogeneity of geographical phenomena(O’Neill et al. 1988, McGarigal and Marks 1994). One of the first attempts to integratesome specific spatial properties in the measure of entropy was suggested by Li andReynolds (1993). The main idea behind this was to quantify a measure of contagion andto which extent regions of a given class are adjacent to regions of another class, allowing toevaluate the degrees of juxtaposition and aggregation of the categorical data. Other mea-sures such as dominance have been also suggested (Riitters et al. 1996). In fact, contagion isinversely correlated to diversity. For a given number of classes, the contagion is minimumwhen all classes are evenly distributed and equally adjacent to each other. As such themeasure is essentially expressing a local configuration tendency, the overall spatial structureof the categorical data considered deriving from the accumulation of local information. Inregional science where areal data with weights has to be taken into account, a density-basedmeasure of entropy has been applied to the study of a probability distribution over aprogressive distance from a given location (Batty 1974), but still, the relative spatialdistribution of the classes is not taken into account. With similar approaches to regionalscience, segregation analysis coming mainly from social science has been using entropymeasures for spatial data (e.g., Wong 2002, Reardon and O’Sullivan 2004). Besides usingdensity-based entropy, spatiality in the entropy measure is taken into account with localmeasures derived using neighborhoods densities of each category (e.g., social groups) usinga spatial proximity matrix, see also Karlstrom and Ceccato (2002). This chosen spatialproximity matrix is also used to derive exposure indices between two groups or one to theothers from a correlation like formula in similar way to a local Moran’s Index (Anselin1995). In this approach each category of the categorical variable studied is reified as onevariable observed on areal data (proportion of each category in each unit), therefore usuallylooking at the spatial structuring of one category or spatial correlation between two groupslike in hot-spot maps (see also Section 3).

The main objective of this paper is to be able to relate the spatial or spatio-temporalpattern of several categories together in one single index as a global value or as a spatialmap expressing local values. The focus of the paper is limited to explore how the conceptand previous measures of entropy can fulfill this objective.

In a previous work, a measure of spatial entropy has been introduced to take intoaccount the role of spatial distances between classes when applying a measure of entropy(Claramunt 2005). The idea behind this notion is to consider the primal role of distancetoward the spatial structure of a given system. The principle of this spatial entropy is thatthis measure should augment when distance between dissimilar entities decreases, as wellas the entropy should augment when the distance between similar entities increases. Assuch this measure gives an overall and global index of the relative distribution in space of

the classes. On the other hand, local arrangements as well as cross-relations between theclasses are not directly evaluated. With this aim a related parallel work on local interac-tions between co-occurrences of categorical data has been studied and formalized intoanother spatial measure of entropy (Leibovici 2009; Leibovici et al. 2011b). The mainidea of the latter indices, suggested as generalizing contagion indices, is to consider asspatial information the distribution of co-occurrences between two or more observationswith varying vicinities for the exploration of spatial patterns at different scales. It appearsthat these two approaches take different topological points of view when extending themeasure of entropy to spatial observations: local versus global, co-occurrence versusrelative spatial distribution. The principle of the co-occurrence-based entropy is verysimilar to the symbolic entropy framework developed in regional science by Lopezet al. (2010) and Matilla-García et al. (2012), nonetheless the occurrence of a symbolbeing defined as matching the nearest neighbors pattern of categories may be quiterestrictive and strict for spatial dependence whereas co-occurrence as defined inLeibovici (2009) brings more flexibility (see Section 2.1).

Whilst the aim of the paper is to extend these approaches under the interrelations ofthe spatial and temporal dimensions, the research developed in this paper explores thecomplementarity of these two families of indices, and evaluates to which degree thecombination of the two might enrich the way a given phenomenon distributed in spaceand time can be analyzed in order to apprehend its spatio-temporal structuring.

Spatial structuring analysis can be seen as the alter ego of spatial clustering analysis,where in the latter the focus is more on locating and delineating clusters often from testinga hypothesis referring to a null distributional assumption. The semantic behind thewording structuring is more related to pattern, modeling, association and to the appre-hending of the spatial observations as a whole from which the existence of clusters is oneaspect among other structuring characteristics. Spatio-temporal clustering from a structur-ing point of view comes from disciplines such as ecology with criteria like diversity,richness, patchiness and associations whereas detecting structures by locating clusters isoften linked to outbreak detection as in epidemiology. Local and global approaches haveproven to be useful in spatial analysis and are also at the source of a plethora of spatialand spatio-temporal methods to do with structuring and tests for clustering (Haining 2003,Tango 2010, Bivand et al. 2013).

Concepts such as composition and configuration echoing global and local properties(Li and Reynolds 1993, Karlström and Ceccato 2002, Boots 2003, Okabe et al. 2010) areat the root of a proper description of the spatial distribution of a categorical variable; theseterms are directly linked to occurrences and co-occurrences. Global configuration can bealso intrinsically expressed by distances between the observations of the classes involved.Ripley’s K function (Ripley 1977), seen as the ratio of the expected number of furtherpoints from a random point location for a given spatial lag (distance) to the local intensity,is a global statistic that describes the structuring which in fact expresses the naturaldependency between composition and configuration when studying the second ordermoment of a spatial point process. The shape of the curve of the observed values for Kat different spatial lags and its comparison to a theoretical curve obtained under particularhypothesis such as complete spatial randomness (CSR) helps to describe the global patternand test it (e.g., Bivand et al. 2013). Cross-K statistics extend the approach to two sets ofpoints with a categorical variable identifying the points. Nonetheless with more than twocategories (classes) using cross-K functions for all pairs of categories (or pooling all othercategories against one) may be limited when describing the global spatial structuring.Higher-order co-occurrences have been proven to be useful at describing more complex

associations including using extensions of Ripley’s K (Leibovici et al. 2011b). Unlikemost of the statistics based on second order moments the entropy allows to grasp allcategories at once. The entropy indices introduced aim at enriching the conventionalmeasures of entropy as identified by the information theory by taking into account thespatial and temporal dimensions structuring the information. Nonetheless our approachmay also benefit the research into spatio-temporal clustering within an outbreak detectionsense as entropy-based testing have also been proven to be as powerful as classicalapproaches for spatial dependence (Matilla-García et al. 2012).

The rest of the paper is organized as follows. Section 2 introduces the measures ofdistance-based and co-occurrence-based spatial entropies, while Section 3 studies the localand global aspects of the indices. Section 4 extends our approach to the temporaldimension and discusses the application potential of the whole framework. Whilst pre-vious sections were illustrated with a simulated dataset, Section 5 describes preliminaryresults obtained for a study of conflicts of maritime activities in the Bay of Brest (France)in 2009. Finally the conclusion summarizes the contribution and outlines further work.

2. Distance-based and co-occurrence-based spatial entropy

Along the paper the principles of the various spatial and spatio-temporal entropy indicesare illustrated using simulated data with 3 classes of 60 points evolving from time 1 totime 6. The purpose of this simulation is to generate and control the spatio-temporalstructure of the data using a known evolution process with simple structuring for eachclass. Evaluating to which degree the methodology described in the paper could pick upthe existence of the deterministic part of the evolution process is beyond the scope of thepaper. Figure 1 shows these three classes generated with different initial distributions andevolutions. At time 1 the classes were generated on a unit square window as: + withrandom uniform coordinates x and y, o idem with a tendency to cluster in the upper rightusing a density proportional to xy and Δ with random uniform coordinates with a tendencyto cluster in the bottom right corner using a density proportional to x2ð1� yÞ2. At each

time 1 time 2 time 3

time 6time 5time 4

Figure 1. Spatio-temporal point pattern simulated with three classes.

new time step the points of class + have Gaussian shifts, the points of class o movelinearly (coefficient 0.2) toward the point in the top right corner of coordinates (0.8, 0.8),and the points of class Δ move toward the horizontal axe proportionally to the verticalcoordinate (with coefficient 0.2).

For each of these six marked point patterns the normalized Shannon entropy (1) of theobserved distribution is equal to 1, as the distribution of marks (classes) is uniform.Nonetheless, the observations of these classes, except the points of class +, clearly exhibita spatial organization, a pattern which becomes more and more evident with time. Theresearch aim of the paper is to develop appropriate spatial and spatio-temporal entropystatistic being able to detect the existence of this structuring that emerge from theseevolution patterns.

As being not just a distribution over a set of spatial units, a spatial distribution shouldreflect the information about the contiguity of observations with dependencies on attributevalues. For example, in Figure 1 even if the count of occurrences of observations is thesame at time 1 and time 6, a spatial entropy at time 6 should be smaller as points are moreorganized for the classes o and Δ. Similarly, permuting randomly the pixels of an imageshould not either give the same entropy. Therefore using solely the frequency distributionof the classes in the entropy index cannot reflect the structures that emerge in space andtime. In order to take into account the spatial or spatio-temporal dimensions within anentropy index, the approach developed aims at integrating two topological and metriccriteria in the definition of the entropy: distances between and within classes, and co-occurrences of observations of the same class or of different classes.

2.1. A co-occurrence-based spatial entropy

The proximity of the occurrences for a particular class is crucial when assessing the spatialdistribution of a categorical variable. The concept of co-occurrence, that is, a set ofobservations located within a given spatial zone, is therefore a characteristic to take intoaccount when describing the spatial distribution. In order to take into account the spatialpatterns that emerge from several classes, a measure of entropy can be defined using theco-occurrences distribution (Leibovici 2009; Leibovici et al. 2011b). This is devised as ageneralization of the adjacency index (O’Neill et al. 1988, Li and Reynolds 1993). A co-occurrence distribution is obtained, in a practical approach, by counting the number ofcollocated sets of k observations (k is called the order of co-occurrence), a collocationtaking place when the distance between any two observations of the set is less than achosen threshold d, the collocation distance. Depending on the attributes of the observa-tions to record for the co-occurrences, the co-occurrence distribution is finally character-ized. The k-spatial entropy of a categorical variable C with a total of Cj j classes observedspatially is a total of N points is defined as

HkSðC; dÞ ¼ �1= logðjCoojÞ�coopcoo;d logðpcoo;dÞ (2)

where pcoo,d is the multinomial distribution associated to the multi-entry table of counts ofco-occurrences of order k at collocation distance d; the multi-index coo relates to thechosen attributes to record, associated to the chosen categorical variable(s).Therefore,jCooj, the number of different values that the multi-index coo for the categoricalvariable C can take, is the number of co-occurrence classes. The normalizing value oflogðjCoojÞ corresponds to the maximum value of the Shannon entropy for a uniformdistribution of these classes, then 0 � HkS � 1. With a multi-index coo chosen as a

repetition of a class according to the order k, e.g., pcoo; d ¼ pccc; d when k = 3 for anyclass c, this entropy measure is termed self-k-spatial entropy Hs

kS :

HskSðC; dÞ ¼ �1= logðjCjÞ�i pccc...c;d logðpccc...c;dÞ (3)

The distribution used in the self-k-spatial entropy comes from keeping only the hyperdia-gonal of the multi-entry table of co-occurrences counts that would be used for the k-spatialentropy: in Equation (3) jCooj ¼ jCj, in Equation (2) jCooj ¼ jCj � jCj � jCj with k = 3.Therefore, Hs

kS the self-k-spatial entropy appears as a multinomial-univariate index versionof HkS, the k-spatial entropy which is multinomial-multivariate. While the classicalentropy is based on the distribution of occurrences of the classes, the self-k-spatial entropytakes into account the distribution of co-occurrences of these classes.

In the rest of the paper the co-occurrence-based entropy uses only the self-k-spatialentropy as in first instance we are interested in trends emerging from the set of classes butnot in the associations of the classes. This choice is driven by a more direct analogy then,for the two approaches of co-occurrences and distance-ratios, of the way the topology ofthe classes is taken into account, thus allowing potential integration of the twoapproaches. The multivariate aspect of the k-spatial entropy is also more demanding onsample sizes (Leibovici et al. 2011b).

In Figure 2 the choice of a range of collocation distances illustrates the sensitivity andpotential scale analysis provided by the self-k-spatial entropy. This shows that, indepen-dently of the evolution, a collocation distance too small may induce a false impression ofpattern structure, while a too large one can also be not appropriate to identify the emergingspatial structure. Regarding the patterns detected, at a local scale the entropy valuesdecrease with time while at a higher scale from a collocation distance d ≥ 0.4, the entropybecomes relatively stable after time 4. That distance d where the observed entropychanges rapidly (between 0.0 and 0.4 here) can be considered as an important thresholdfor the analysis of the scale of the studied data. Moreover, using a range of collocation

time 1 time 2

collocationd = 0.11d = 0.22d = 0.33d = 0.44d = 0.55

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Figure 2. Evolution over time of the self-k-spatial entropy with different collocation distances.

distances allows consistent pattern detection, here identified from the simulated data. Theplateau of entropy values, observed from d ≥ 0.4 in Figure 2 illustrates the ‘maximum’scale structure which can be detected but up to d = 0.3 denser clustering structures areidentified up to the end of the evolution.

2.2. Discriminant-ratio-based spatial entropy

The distances between the observed classes can also reveal the ‘spatial correlation’ of theclasses. Intuitively, if the observations for a particular class tend to cluster relatively to theother classes, the distribution of distances between pairs of observations from that classshould be narrower and shifted toward small values than the distribution of pairs ofobservations with one observation not in that class. For a categorical variable C with adistribution of observations pc ¼ nc=N (nc being the number of observations of the classc and N the total number of observations), a measure of spatial diversity has beenintroduced using a ratio distances between pairs of observations for similar and differentclasses (Claramunt 2005):

HsðCÞ ¼ ��cdintc

dextc

pc logðpcÞ (4)

where dintc ¼ 1=ð2 � nfc;cgÞP

r2c; r02c drr0 is the average distance between a pair of obser-vations belonging to the same class c, called intra-distance;dextc ¼ 1=nfc;c0�cg

Pr2c; r02c0 drr0 is the average distance between a pair of observations,

where only one belongs to the class c, called the extra-distance (the number of pairs are,respectively, nfc;cg ¼ ncðnc � 1Þ=2 and nfc;c0�cg ¼ ncðN � ncÞ). The ratio of these dis-tances appears as a discriminant statistic between classes with compact observationsrelatively to the other classes, giving a ratio smaller than 1, and classes more dispersedrelatively to the others giving a ratio greater than 1.

Taking the inverse of this ratio one gets, after normalization, a probability with asimilar aim as with the co-occurrence distribution (high probability for compact clusteredobservations of a given class):

HdSðCÞ ¼ �1= logðjCjÞ�c dρc logðdρc Þ (5)

where dρc ¼ ðdextcdintcÞ=ðPc

dextcdintcÞ is the normalized ratio of the distances. HdS is easier to

interpret than HS but one loses the relative distribution of observations of the classes,which can be re-integrated by doing the symmetric of Equation (4):

HpdSðCÞ ¼ �jCj= logðjCjÞ�c pcd

ρc logðdρc Þ (6)

Or

HpdSðCÞ ¼ �jCj= logðjCj2Þ�c pcdρc logðpcdρcÞ (7)

which can be seen as the normalized entropy for the diagonal of the joint distribution ofthe observations of the classes and their compactness under the hypothesis ofindependence.

When comparing the values of the different entropy indices so far introduced, andaccording to their evolution for this particular simulated dataset, the discriminant-ratio-based entropy HS appears to be the most responsive to the spatial structuring and itsevolution, nonetheless in a ‘linear’ way relatively to the evolution. The self-k-spatialentropy index Hs

kS , at medium collocation distances, performs well whilst being more ableto show the acceleration of the spatial structuring. The new indices based on thediscriminant-ratio, HdS, HpdS and Hp

dS are less able to capture the dramatic changes inthe evolution. One must notice that because of the uniform distribution of the classes(globally), the ‘entropy part’ of the index HS is the same for all the classes, so does notinfluence the changes (Figure 3). The index works only on the discriminant-ratios in thiscase.

3. Local and global indices

The spatial entropy indices developed in the previous section are global statistics that shouldbe completed by local indices that can identify where (or when or both) the spatial or timestructure is the most determinant. Particularly low and high values along with their auto-correlation and potential grouping may be then looked for and a post-hoc testing analysis maybe then applied to estimate their significance (Leibovici et al. 2011a). Complementing globalstatistics by their local equivalent has been studied by spatial correlation measures ingeostatistics and spatial analysis with applications in geographical information science(GIS). One can mention LISA statistics (Anselin 1995) with the Moran’s Index, or theGetis–Ord statistic (Getis and Ord 1992), which are frequently used for continuous variablesrepresented by ‘hot-spot’ maps, thus could be used for post-hoc analysis. When consideringcategorical data, scan statistics have been also applied using for example SaTScan and GAMmethods (Openshaw et al. 1987, Kulldorff and Nagarwalla 1995), and where a windowaround a set of observations is applied using a maximum likelihood ratio statistic thatcompares inner and outer windows: the set of observations form a cluster if the statistic issignificant enough. As described in Leibovici et al. (2011a), a SaTScan method for

time 2baseline

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P Hpdsp HdSpC HS

HdsR

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d = 0.11HkS HdSs HS

d = 0.22d = 0.33d = 0.44

Figure 3. Comparison of the evolution of the ratio to baseline of the entropy indices: HkS is theself-k-spatial entropy (3), HdSs (HdS, HpdS, HdSp) and HS are the distance-based indices inEquations (5), (7), (6) and (4), respectively.

multinomial data has been developed by Jung et al. (2010) with as alternative hypothesis thatthe proportions for at least one of the categories show different values between the inner andouter windows of the scan, so not necessarily pointing out a particular distributional structureof the categories in the inner window.

3.1. Local contributions to global statistics

While LISA’s statistic properties assure that the sum of the values is proportional to theglobal statistic, this is difficult to achieve here because of the use of logarithms in theindices introduced in the previous sections. Nonetheless, co-occurrences and discriminantdistances-ratios, accumulated from local observations, can provide meaningful localevaluations that can also give some insights regarding the global distribution. For thediscriminant-ratio-based entropy the ratio of the averages of the distances inter and extracan give a local evaluation:

dinti

dextiðViÞ ¼ ðnextðViÞ=nintðViÞÞ�r2cðiÞ˙Vi

dri=�r‚cðiÞ;r2Vidri (8)

for a neighborhood Vi for each observation i, (next(Vi) and nint(Vi) being the count ofobservations for each sum). Equation (8) becomes (4) if the vicinity rule is, Vi = V, i.e., thewhole study area. The neighborhood Vi can be defined (a) from a zoning system (e.g.,administrative or naturally defined units) where each Vi matches one of the fixed unitscontaining the observation i or (b) from an algorithmic constraint where Vi is a bufferaround each observation i using a buffer distance dependent or not from i or (c) where Viis a varying buffer chosen according to sample size using nearest neighbors. Overall, thevicinity rule allows expressing a local characteristic at each observation.

In Figure 4 a map of the discriminant-ratios computed at each point using (8) showsthe overall influence of the observations from all classes to a particular (localized) classelement: a small value reveals a tendency to be in a cluster of the observed class. InFigure 4 the values get smaller with the evolution, and they are even smaller for the classo, confirming the spatial structure.

Results in Figure 5, with Vi being not the whole space and constituting a local vicinityof the 90 nearest neighbors for each observation i, are comparable but with less evidenceof clustering for less compact clusters, e.g., at time 4. This seems logical as the distancesare then limited within the local neighborhood. Nonetheless local clustering for a non-globally clustered class is better depicted than when using the whole space. This issomehow confirmed when comparing the time 6 between Figures 4 and 5, where theclass + exhibits visually a zone in between the other two classes, seen as a structuringcluster by the lack of observations from other classes: small values in Figure 5 but not inFigure 4.

Regarding the self-k-spatial entropy, the local contributions to the global statisticemerge from the local co-occurrence counts for the observed class, i.e., co-occurrenceswith the current observation as illustrated in Figure 6. A high value means a tendency tobe in a cluster of the observed class, and the emerging clusters are depicted by theevolutions of these local contributions.

In Figure 6 the neighborhood is defined by the collocation vicinity. Using a largerneighborhood Vi allows to consider local evaluation or local constraints instead of a local

contribution. In Figures 4 and 5 a scaling was applied so that the sum of all values overspace and time are similar.

3.2. Local evaluations and local constraints

The introduction of a neighborhood parameter for the local contributions to the spatialentropy (using either a distance-ratio-based index or a k-co-occurrence one) allows alocalization of the indices. Nonetheless, there is a difference between the two approaches:the co-occurrences-based indices being subject to the collocation distance and expressing

time 1 time 2 time 3

time 6time 5time 4

Figure 4. Map of the discriminant-ratios at each point Equation (8) with Vi as the whole space.Sizes of labels are proportional to the values (longitudinally normalized).

time 1 time 2 time 3

time 6time 5time 4

Figure 5. Map of the discriminant-ratios at each point Equation (8) with Vi as a set of the 90nearest neighbors. Sizes of labels are proportional to the values (longitudinally normalized).

a scale effect are already locally focused. The co-occurrence distribution is an accumula-tion of counts but for the distance-based entropy indices, the discriminant-ratio is a globalstatistic as computed using all the observations available. Nonetheless, the distance-basedentropy indices can be evaluated locally in the neighborhood of each observation or canbe also evaluated globally but with a vicinity constraint when averaging the distances ofpairs:

dintc

dextc

ðVÞ ¼ ðnextðVÞ=nintðVÞÞ�r�r02cðrÞ˙Vrdrr0=�r�r0‚cðrÞ;r02Vr

drr0 (9)

then leading to new versions of the discriminant-based spatial entropy indices of theEquations ((4)–(7)): HSðC;VÞ; HdSðC;VÞ; Hp

dSðC;VÞ; HpdSðC;VÞ, where the vicinity ruleleading to each Vr is expressed by V. The constraint expressed by the vicinity rule V is asdescribed in the previous section, using either a fixed zoning constraint (the sets of all theneighborhoods of each observation defined by a buffer distance) or the number of nearestneighbors. In Figure 7, 90 nearest neighbors were used; the self-k-spatial entropy resultsfrom Figure 3 are repeated as a benchmark. In comparison to Figure 3 the index HS isdiscriminating less with the 90 neighbors constraint than without, but is still within thebest results obtained for the self-k-spatial entropy with collocation distances between 0.1–0.3. The other distance-based entropy indices do not capture the clustering evolution ofthese three classes.

In Figure 8, the effect of different neighborhood sizes is analyzed on the indices fromEquations (5) and (4), HdSðC; VÞ; HSðC; VÞ, respectively, and updated with the neighbor-hood constraint (9). The general emerging pattern is that the higher the numbers ofneighbors, the more the indices capture the clustering evolution. Nonetheless, for thetwo indices 90 neighbors seems to be a tipping point from where the indices get their‘power’, this is clearly visible for HSðC; VÞ bottom of Figure 8 but also apparent on theless powerful index HdSðC; VÞ. Interestingly, and relatively to the baseline (time 1),

time 1 time 2 time 3

time 6time 5time 4

Figure 6. Map of the self-k-co-occurrence counts at each point with collocation distance d = 0.3.Sizes of labels are proportional to the counts.

HSðC; VÞ is less discriminant for the evolution with all the observations (no neighborhoodconstraint) than with the 150 nearest neighbors constraint.

The same vicinity rule can be also applied for local evaluations but the sample sizeshave their importance. This approach has been used (Leibovici et al. 2011a) for a spatialscan statistic where the number of points considered for each local evaluation is given bythe number of nearest neighbors within a ‘hot-spot’ mapping approach. It can be appliedfor both spatial entropy approaches taken here.

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times 2bbaseline times 3 times 4 times 5 s 6times

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Figure 8. Effect of the size of the vicinity size of the vicinity constraint for HdSðC; VÞ; HSðC; VÞ.

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% o

f bas

elin

e en

trop

y

d = 0.11HkS HdSs HS

d = 0.22d = 0.33d = 0.44

Figure 7. Comparison of the evolution of the ratio to baseline of the entropy indices forHSðC; VÞ; HdSðC; VÞ; Hp

dSðC; VÞ; HpdSðC; VÞ where the vicinity constraint is defined by 90 near-est neighbors at each location.

In Figure 9, despite a decrease between time 1 and time 6, not much variation occurs:HsðCjVÞ shows more time variations while HdsðCjVÞ shows more spatial variations.Constraining too much the extra-distance dextc to local observations makes the distance-ratio less discriminant. The self-k-spatial entropy evaluated locally works much better(Figure 10), but that was expected as the global statistic is based on aggregation of localcounts of co-occurrences.

Co-occurrence-based indices being more locally dependent, the values displayed inFigure 10 highlight logically the local structuring and autocorrelation of this structuring.The local evaluations compare observed co-occurrences distributions to a local uniformityof co-occurrences, which can be seen as a local null hypothesis. The control within thisstatistic can be qualified as internal as varying the size of the scan window do not changethe null hypothesis expected counterparts, unlike the principles used with SaTScancomparing inner and outer log likelihood (KullDorff and Nagarwalla 1995) or with

time 1 time 2 time 3

time 6time 5time 4

time 1 time 2 time 3

time 6time 5time 4

Figure 9. Local evaluations termed HsðCjVÞ (top) and HdsðCjVÞ (bottom) where V is a 90neighbors scan window for each local spatial evaluation.

Okabe et al. (2010)’s approach where within homogeneity assumption the local observedcounts are compared to a binomial valued distribution of the expected counts. The indicesbased on the distance-ratios also have an internal control as the statistics do not rely onouter estimations or external hypothesis, nonetheless as seen in Figure 8 its discriminantpower increases with the size of the local neighborhood used.

4. Spatio-temporal integrated approach

Extending a spatial analysis methodology to the spatio-temporal domain may not be asstraightforward as it could seem and the literature in spatio-temporal clustering, trajectoryanalysis, spatio-temporal GIS is very active (Rinzivillo et al. 2008, Kuhn 2012, Bivandet al. 2013, Gabriel et al. 2013). In a previous work the spatial index given in Equation (3)has been extended to a spatio-temporal index as follows (Claramunt 2012):

HST ðCÞ ¼ ��cdSTintc

dSTextc

pc logðpcÞ (10)

where the intra-distances (respectively, extra-distances) take into account the spatio-temporal dimension, and is modeled as the product of the intra-distances (respectively,extra-distances) in space and time:

dSTintc ¼ dTintc dSintc (11)

A similar multiplicative approach has been taken by Knox (1964) and particularly Mantel(1967) who built a statistic to test for spatio-temporal clustering based on the followingstatistic:

dSTintc ¼ Z ¼ 1=ð2 � nfc;cgÞ�r2c;r02cdTrr0dSrr0 (12)

time 1 time 2 time 3

time 6time 5time 4

Figure 10. Local evaluations for the self-k-spatial entropy with V is a 90 neighbors scan windowfor each local spatial evaluation with collocation distance 0.3.

which used standardized distance values (centered and divided by standard deviations ofthe distances in space separately in time and space). Two different spatio-temporal entropyindices (10) according either to using the macro-spatio-temporal distance model (11) orthe Knox–Mantel or KM spatio-temporal distance model (12) can be derived and the samefor the spatio-temporal versions of (5, 6, 7).

Looking at co-occurrences and in reference to the Knox approach, the distances in(12) are now co-occurrence functions of order 2 here: oTrr0 ðdÞ ¼ 1 if dTrr0 � d, with d thecollocation distance (called the critical time distance by Knox) and idem for space, thenleading (instead of dSTintc ) to a co-occurrence spatio-temporal model oSTintc :

oSTintc ðdS; dTÞ ¼ 1=2�r2c;r02c oTrr0 ðdTÞoSrr0 ðdSÞ (13)

counting the number of pairs of observations from the same class that are co-occurrent inspace and in time, and a similar definition for oSTextc . A similar co-occurrence approachwith the model as in (11) will still build a suitable spatio-temporal compactness ratio. TheKnox approach translates directly for the co-occurrence-based indices of Section 2.1 byusing as in (13) the collocation distances dT and dS; indices have a superscript o, e.g.,Ho

kST ðCÞ. For the other distance-ratios-based indices expressed in Section 2.2 with theirspatio-temporal versions, alike (10) and with (11) or (12), the Knox approach using onlyco-occurrent counts instead of distance averages makes these indices more local as thecollocation distances in time and space parameterize them. Nonetheless, as distances andco-occurrences are ‘negatively correlated’, the inverse of the ratio has to be considered in(4) and (8) as was done in (5). These distance-ratios indices, termed with also the addedsuperscript o, e.g., Ho

ST ðCÞ, combine the discriminant-ratios approach and the co-occur-rence approach (here of order 2). For them the model expressed by Equation (13) is in factequivalent to count the co-occurrences of pairs within a spatio-temporal vicinity similar toa ‘cylinder’ selection (the high of the cylinder being the collocation distance in time)before computing the ratios.

Local constraints and local evaluation as in the previous section can be combined withthese type of discriminant-ratio indices. Table 1 illustrates some results for the simulateddata; as it might be difficult to assess the results for all times (except with the normalizedindex in column three), a sliding window selection has been applied with comparison toall times.

In Figure 11, the global evaluation for the self-k-spatial entropy HsokST ðC; dS; dTÞ of

80.5% has been locally evaluated at time 3 and 4, and shows a clear visual spatio-

Table 1. Global spatio-temporal entropy for different time selection. (the equation numbersinvolved for the indices are mentioned).

Time\indexvalue (% of{all times})

HmacroST ðCÞ

(10) and (11)HKM

ST ðCÞ(10) and (12)

HodST ðC; dT ; dSÞ(5) and (13)*

{time 1, time 2, time 3} 1.19756 (113%) 1.20049 (111%) 92.9% (107%){time 2, time 3, time 4} 1.08373 (103%) 1.08513 (100%) 88.2% (101%){time 3, time 4, time 5} 0.98823 (93.2%) 0.98804 (91.4%) 83.7% (96.2%){time 4, time 5, time 6} 0.91353 (86.1%) 0.91298 (84.5%) 81.0% (93.0%){time 1, time 3, time 5} 1.11386 (105%) 1.13269 (104.8%) 88.4% (102%){all times} 1.06072 (100%) 1.08033 (100%) 87.0% (100%)

Note: *critical collocation distances: time 1, space 0.3.

temporal clustering (small values are to look for). This pattern detection is less pro-nounced when using the local entropy representation with Ho

dST ðC; dT ; dSjVÞ with aglobal value of 87% (Table 1) and with the local punctual contributions to HKM

ST ðCÞ(where also small values are expected when clustering structures occur). ComparingFigure 11 and previous figures on local contributions and local evaluations where onlythe spatial component was used, the integration of the temporal dimension brings anadditional variation whatever the choice of the spatio-temporal model.

Using a macro-multiplicative or micro-multiplicative approach (KM), several differentways of integrating the spatial and temporal dimensions in the indices have been derived.These main approaches can lead to other combinations such as using the macro-modelwith co-occurrence-based indices by performing the Hadamard product (Styan 1973) ofthe spatial and temporal co-occurrence tables as the observed spatio-temporal co-occur-rences. Combining a local constraining and a macro or KM space-time model bringsanother four indices based on distance-ratios purely, and two others combining distance-ratios and co-occurrences when using the Knox approach, as with Ho

ST ðCÞ previously.Finally time and space can be integrated using the above series of models and applicationsfor the entropy approaches with co-occurrences or distance-ratios but now with a non-separate approach. This concerns local evaluations or local constraints but also the Knoxapproach in (13). So far being found close in space had no influence of being found closein time but it seems usually more meaningful to look further in time when locations areclose in space and vice versa. This non-separable approach means for example that thespatio-temporal cylinder neighborhood around a point becomes more biconic as thespatial diskal section becomes smaller as one goes further apart in time similarly to anoutbreak model (Tango 2010).

Altogether and depending on the use of global, locally constrained or local indices,using the spatio-temporal entropy indices presented in the paper means choosing two to

contributions time 3

contributions time 4

HdST time 3

HdST time 4

HkST time 3

HkST time 4

Figure 11. Local contributions to the HKMST ðCÞ (left), local evaluations of the self-k-spatial entropy

HsokST ðC; dS; dT jV Þ (right) and of the discriminant-ratio spatial entropy Ho

dST ðC; dT ; dSjV Þ (middle)at time 3 and time 4 (over the whole spatio-temporal domain with collocation distances 1 in timeand 0.3 in space): local evaluations defined by V the 90 nearest neighbors in space and distance 1 intime.

four parameters (2 for collocation, 2 for neighborhoods) but none for distance-ratiosentropy as global index unless a local constraint is used. Applying a local constraint issimilar to performing a local evaluation but accumulating the results for a global index.Fixing the values chosen for these parameters should be driven by the case study itself andthe application requirements attached to it. Besides sample size issues when choosing forexample a spatial collocation distance too small, exploring a range of parameters is part ofthe methodology in order to be able to describe scale variations aspects. Sensitivityanalyses may be useful when reporting a specific range of parameters linked to aparticular interpretation.

5. Case study

The application of the entropy indices on the simulated data illustrates their potential.Further experiments have been applied to a research currently under development andoriented to the study of the spatio-temporal distribution of maritime activities taking placein the Bay of Brest in North West France during the year 2009 (Le Guyader andGourmelon 2013). This research takes place in the context of coastal seas, in whichdiverse activities take place, this generating an increasing pressure on the environment andoften conflicting interactions (Young et al. 2007). Understanding these interactionsremains a challenge for research (Leslie and McLeod 2007).

The identification of activity conflicts at sea can be modeled by superimposingactivity zones (Brody et al. 2006, Beck et al. 2009, Stelzenmüller et al. 2013), andquantified using several measures of spatial intersections such as the cumulative numberof activities, activity density per unit of area, presence/absence of potential conflicts ordegree of potential conflict. However, the temporal dynamics of these activities are notconsidered, as well as it is not straightforward to qualify the way these activities interact inspace and time. We formulate two hypotheses: (1) activities potentially interacting areconsidered in spatio-temporal interaction; (2) spatio-temporal interactions are approxi-mated by their intersections in space and time.

Daily human activity patterns are recorded over a period of one year in 2009, and bytheir spatial, temporal, quantitative and qualitative properties. Data are collected fromAutomatic Identification System (AIS) that track maritime trajectories, and semi-struc-tured interviews realized with stakeholders. Overall 29 activities have been recorded (e.g.,fishing, water sports, maritime transportation) with a daily temporal granularity. Dailyspatio-temporal intersections have been discretized with a uniform (hexagonal) lattice andaggregated monthly. For the purpose of our study spatio-temporal interactions betweenmaritime transportation activities (labeled C111) and leisure have been considered, that is,

AtlanticOcean

France IroiseSea

kayak (E115), windsurf (E113) and sailing (E111), this generating six classes of conflict inspace and time: (E111_ C111), (E111_E113), (E111_E115), (E113_C111), (E113_E115)and (E115_C111). Each spatio-temporal conflict is approximated in space by the centroidof the interaction polygons that model the intersection of the polygons where each activitytakes place.

A first visual exploration (Figure 12) shows that the classes appear well delineated inspace at any month. Therefore, the analysis is oriented to a study of spatio-temporalvariations in order to explore the persistence or not of the conflicts of activities.

Among the entropy indices previously introduced the firsts results illustrates the useof HKM

ST ðCÞ, HodST ðC; dT ; dSjV and Hso

kST ðC; dS; dY jVÞ for overall, local contributions and

E115_C111E113_C115E113_C111E111_E115E111_E113E111_C111

Sept

FebJanDecNovOct

AugJulJunMayAprMarall months

Figure 12. Visual representation of conflict classes overlapping over the months.

Jan Feb Mar

Apr May Jun

Jul Aug Sept

DecNovOct

Figure 13. Densities of the local contributions to the HKMST ðCÞ enabling to identify a potential

spatio-temporal structure.

local evaluations where C is the categorical variable of the six classes of conflictsidentified for this dataset.

Figure 13 shows that nonetheless the classes appear quite clustered in space whateverthe month, they show some variations in term of spatio-temporal structuring. We focusedin the neighborhood of the month of August as its density appears well structured in twocomparable groups of high and low entropies (contributions to) and as being a month ofhigh activity in general. The local evaluations allow describing and interpreting the resultper classes but the indices computed locally take into account all the classes. In Figure 14some classes like E111-C111 or E113_E115 have a tendency to have higher contributionsto the spatial structure that emerges and some like E115_C111 or E113_C111 have asignificant variation due to time. This observed statement from the extent and compact-ness of the classes observations with relatively high scores were confirmed by computingcorresponding either spatial or temporal indices. Note the class E111-E113 (in red) hasvery homogeneous values and to a lesser extent this is also the case for class E111_E115.These two classes have shown an increase in their corresponding term in the sum makingthe distance-ratio-based entropy, from its computing with only spatial distances ðHsðCÞÞto when using spatial and temporal distances and where the Knox–Mantel model was

N = 460E111_C111

N = 551E111_E113

N = 460E111_E115

N = 400E113_E115

N = 1900E113_C111

N = 93E115_C111

Figure 14. Local contributions at month August to the HKMST ðCÞ sizes proportional to the inverse of

the score (high contribution here). Split display of the six categories composing the variable C.

applied ðHKMST ðCÞÞ. This expresses a lack of temporal persistence of the potential cluster-

ing for these two classes.Figure 15 depicting local entropies for the Ho

dST index, concludes similarly to as forFigure 14 concerning the class E111_C111, nonetheless with refined clusters. The spatio-temporal structuring identified for class E113_C111 in Figure 14 are conclusive but nowE111_E113 shows strong inhomogeneities. The results shown for the indexHso

kST ðC; dS; dY jVÞ in Figure 16 confirm previous results altogether; the spreads of valuesappears more discriminative for the clustering zones previously identified, at similarlocations but with different shapes. Figure 16 exhibits more evidence for clustering inareas where the logic ‘cross-roads’ and intense activities is verified from local knowledge.

6. Discussion and conclusion

Overall most of the spatial entropy indices presented in this paper and derived from aproximity concept using distances or co-occurrences, can be either mapped toward similarindices in time or extended to the spatio-temporal domain. Considering global, local andlocally constrained versions of these indices, not only hints the integration of both timeand space in them but in the first place gives much more flexibility to the measure of aspatio-temporal entropy.

N = 460E111_C111 N = 1900E113_C111

N = 551E111_E113

N = 460E111_E115

N = 400E113_E115

N = 93E115_C111

Figure 15. Local evaluations at month August of the HodST (C, dT = 330 m, dS = 2 months jV)

sizes proportional to the inverse of the score (high value for low entropy) and V is defined by theconstraints dT and dS. Split display of the six categories composing the variable C.

The integration of space and time in a distance-based approach can be confusing asthese dimensions do not behave in the same way and scaling may not be alwaysappropriate. Nonetheless, on the simulated data example, the macro-multiplicative andthe Knox–Mantel approaches did not appear to differ much. When considering co-occurrences, the collocation distances (critical distances) allow operating in a separableway whilst still being an integrated approach but can also be thought within a space-timedependency. Besides using co-occurrences at orders k greater than two, non-linear andnon-homogeneous way of defining co-occurrences could be done by replacing a distanceof collocation by a nearest neighbor constraint (Jacquez 1996), e.g., a co-occurrence oforder 3 is counted if each of three observations is part of the nth nearest neighbors of eachother. This is currently investigated by the authors and would allow proximity to bedefined as a local concept.

Novel opportunities regarding the range of phenomena to analyze and the way ofrepresenting the results are enabled by varying the different choices and parameters,particularly the collocation distances, the neighboring and the different ratios used.Overall, the framework identified gives a high degree of flexibility to the whole approachand a potential ability of adaptation to a wide range of applications in the environmentaland urban domains. In previous work the measure of spatial entropy has been alreadyapplied to the classification of agricultural and land-use data in China (Li and Claramunt

N = 460E111_C111 N = 1900E113_C111

N = 551E111_E113

N = 460E111_E115

N = 400E113_E115

N = 93E115_C111

Figure 16. Local evaluations at month August of the HsokST ðC; dS ¼ 44m; dT ¼ 1monthjV Þ sizes

proportional to the inverse of the score (high value for low entropy) and is defined by 250 spatialneighbors within 2 months.

2006), in epidemiology for outbreak detection of spatial association of risk factors(Leibovici et al. 2011a), in ecology for plant communities characterizations (Leiboviciet al. 2011b) and in census data population dynamics (Leibovici and Birkin 2013) whereareal data was considered. Nonetheless most of these examples were mainly focusing onglobal assessments and were dealing with time in a non-integrated way. Post-hoc analyses,such as Monte Carlo envelopes that are applicable here, were used in order to assess thesignificance of the results, bringing together the structuring and clustering points of view.Computationally, distance-ratios are quite efficient as being of order two but co-occur-rences of order three used here may increase computational costs for large sample sizes(spatially and/or temporally; typical run with co-occurrences was quasi instantaneous forthe simulated data − 180 × 6, but was 3mn for the maritime conflict data − 2554 × 12).The functions used in this paper are currently in the process of being packaged as an Rpackage, with more computational costs results.

The spatio-temporal integration along with local exploration (local contributions, localevaluations and local constraints) as proposed in this paper offer now much morepossibilities to apprehend spatio-temporal phenomena measured using an entropy frame-work. We were able to illustrate some of these possibilities with a simulated data and forthe analysis of conflicts of activities in a maritime region in a coastal and local region.From the distributions of the occurrences and co-occurrences of actual and potentialeconomic, social, touristic and transportation activities in a given maritime region underpressure and using the approach described in this paper enable analyzing the differentinteractions that arise in space and time. Amongst the dimensions to explore, the analysisof the conflicts at different scales in space, evolution of the patterns that emerge atdifferent periods of the year as well as the respective influence of the spatial and temporaldimensions are the main aspects to explore. The first results described in the last section ofthis paper confirmed the potential of the different indices toward the analysis of differentspatio-temporal patterns. In this example, where the classes showed visually great spatialclustering in the first place, the temporal persistence of subclusters was detected. Thisallows identifying much smaller clusters enabling better management and understandingof the functioning of the Bay of Brest.

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